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人気のある三角関数 >

証明する 1/(1-tan^2(θ))+1/(1-cot^2(θ))=1

解

証明する

\frac{1}{1 - tan ^{2} ( θ )} + \frac{1}{1 - cot ^{2} ( θ )} = 1

解

真

解答ステップ

\frac{1}{1 - tan ^{2} ( θ )} + \frac{1}{1 - cot ^{2} ( θ )} = 1

左側を操作する

\frac{1}{1 - tan ^{2} ( θ )} + \frac{1}{1 - cot ^{2} ( θ )}

サイン, コサインで表わす

\frac{1}{1 - cot ^{2} ( θ )} + \frac{1}{1 - tan ^{2} ( θ )}

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}} + \frac{1}{1 - tan ^{2} ( θ )}

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}} + \frac{1}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

\frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}} + \frac{1}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}} = 1

\frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}} + \frac{1}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

\frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}} = \frac{sin ^{2} ( θ )}{sin ^{2} ( θ ) - cos ^{2} ( θ )}

\frac{1}{1 - ( \frac{c o s ( θ )}{s i n ( θ )} ) ^{2}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1}{1 - \frac{c o s ^{2} ( θ )}{s i n ^{2} ( θ )}}

結合

1 - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} : \frac{sin ^{2} ( θ ) - cos ^{2} ( θ )}{sin ^{2} ( θ )}

1 - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

元を分数に変換する:

1 = \frac{1 sin ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{1 \cdot sin ^{2} ( θ )}{sin ^{2} ( θ )} - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot sin ^{2} ( θ ) - cos ^{2} ( θ )}{sin ^{2} ( θ )}

乗算:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

= \frac{sin ^{2} ( θ ) - cos ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{1}{\frac{s i n ^{2} ( θ ) - c o s ^{2} ( θ )}{s i n ^{2} ( θ )}}

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sin ^{2} ( θ )}{sin ^{2} ( θ ) - cos ^{2} ( θ )}

\frac{1}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}} = \frac{cos ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

\frac{1}{1 - ( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1}{1 - \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

結合

1 - \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} : \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{cos ^{2} ( θ )}

1 - \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

元を分数に変換する:

1 = \frac{1 cos ^{2} ( θ )}{cos ^{2} ( θ )}

= \frac{1 \cdot cos ^{2} ( θ )}{cos ^{2} ( θ )} - \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot cos ^{2} ( θ ) - sin ^{2} ( θ )}{cos ^{2} ( θ )}

乗算:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{cos ^{2} ( θ )}

= \frac{1}{\frac{c o s ^{2} ( θ ) - s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

分数の規則を適用する:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{cos ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

= \frac{sin ^{2} ( θ )}{sin ^{2} ( θ ) - cos ^{2} ( θ )} + \frac{cos ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

因数

sin^{2} (θ) - cos^{2} (θ) : (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

sin^{2} (θ) - cos^{2} (θ)

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (θ) - cos^{2} (θ) = (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

= (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

= \frac{sin ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} + \frac{cos ^{2} ( θ )}{cos ^{2} ( θ ) - sin ^{2} ( θ )}

因数

cos^{2} (θ) - sin^{2} (θ) : (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

cos^{2} (θ) - sin^{2} (θ)

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (θ) - sin^{2} (θ) = (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= \frac{sin ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} + \frac{cos ^{2} ( θ )}{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}

以下の最小公倍数:

(sin (θ) + cos (θ)) (sin (θ) - cos (θ)), (cos (θ) + sin (θ)) (cos (θ) - sin (θ)) : - (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

(sin (θ) + cos (θ)) (sin (θ) - cos (θ)), (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

最小公倍数 (LCM)

cos (θ) - sin (θ) = - (sin (θ) - cos (θ))

= (sin (θ) + cos (θ)) (sin (θ) - cos (θ)), - (cos (θ) + sin (θ)) (sin (θ) - cos (θ))

(sin (θ) + cos (θ)) (sin (θ) - cos (θ))

または以下のいずれかに現れる因数で構成された式を計算する:

(cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= - (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

- (sin (θ) + cos (θ)) (sin (θ) - cos (θ))

\frac{sin ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

の場合

:

分母と分子に以下を乗じる:

- 1

\frac{sin ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} = \frac{sin ^{2} ( θ ) ( - 1 )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) ) ( - 1 )} = \frac{- sin ^{2} ( θ )}{- ( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

= \frac{- sin ^{2} ( θ )}{- ( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} + \frac{cos ^{2} ( θ )}{- ( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- sin ^{2} ( θ ) + cos ^{2} ( θ )}{- ( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

分数の規則を適用する:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- sin ^{2} ( θ ) + cos ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

キャンセル

\frac{- sin ^{2} ( θ ) + cos ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} : - 1

\frac{- sin ^{2} ( θ ) + cos ^{2} ( θ )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (θ) - sin^{2} (θ) = (cos (θ) + sin (θ)) (cos (θ) - sin (θ))

= \frac{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

キャンセル

\frac{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )} : - 1

\frac{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}{( sin ( θ ) + cos ( θ ) ) ( sin ( θ ) - cos ( θ ) )}

sin (θ) - cos (θ) = - (cos (θ) - sin (θ))

= \frac{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}{- ( sin ( θ ) + cos ( θ ) ) ( cos ( θ ) - sin ( θ ) )}

改良

= - \frac{( cos ( θ ) + sin ( θ ) ) ( cos ( θ ) - sin ( θ ) )}{( sin ( θ ) + cos ( θ ) ) ( cos ( θ ) - sin ( θ ) )}

共通因数を約分する:

cos (θ) + sin (θ)

= - \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) - sin ( θ )}

共通因数を約分する:

cos (θ) - sin (θ)

= - 1

= - 1

= - (- 1)

規則を適用

- (- a) = a

= 1

= 1

= 1

両辺を同じ形式にできることを証明した

\Rightarrow 真

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